Building Simple Words: Multiplication

Multiplication is the process of multiple additions. Multiplication is shorthand for multiple addition operations. Like addition, multiplication is a binary operation, meaning there are two terms. The first number is called the multiplicand and the second is called the multiplier. The result is called the product. The numbers that make up a product are also called the factors.

Multiplication is the process of adding multiple times. For example,
2 × 3 would be the same as 2+2+2=6.

Multiplication can be performed on the number line by starting at 0 and skipping to the right to the answer. Multiplying 2 by 3 would require starting at 0 then skipping to the right by 2's three times. The first skip lands on 2, the second skip lands on 4 and the third skip lands on 6.

0--1--2--3--4--5--6--7--8--9--10->

Just as we can develop a table for addition, we can develop a table for multiplication.

×	0	1	2	3	4	5	6	7	8	9	10
0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10
2	0	2	4	6	8	10	12	14	16	18	20
3	0	3	6	9	12	15	18	21	24	27	30
4	0	4	8	12	16	20	24	28	32	36	40
5	0	5	10	15	20	25	30	35	40	45	50
6	0	6	12	18	24	30	36	42	48	54	60
7	0	7	14	21	28	35	42	49	56	63	70
8	0	8	16	24	32	40	48	56	64	72	80
9	0	9	18	27	36	45	54	63	72	81	90
10	0	10	20	30	40	50	60	70	80	90	100

To find the product of two numbers, we start with the multiplicand at the top and read down the column to the row starting with the multiplier. Where the two meet is the product.

It is useful to memorize the multiplication table, just as it is useful to memorize the addition table.

Multiplying large numbers requires multiplication by each column and then addition of those multiples.

	1	3	5	7
×		2	4	6

We start by multiplying 1357 by 6. Any multiple that is greater than 9 carries the tens digit to the next column to the left as an add after the multiplication of that column. The colored number will be the carry.
7 × 6=42
We keep the ones value and carry the tens value to the next column.
5 × 6=30+4=34
We keep the ones value and carry the tens value to the next column.
3 × 6=18+3=21
We keep the ones value and carry the tens value to the next column.
1 × 6=6+2=8
So our first row of multiplication is 8142.

	1	3	5	7
×		2	4	6
	8	1	4	2

Now we multiply 1357 by 40. Since we are multiplying by a multiple of 10, we place a 0 in the ones column to hold the position open, then simply multiply by 4.
7 × 4=28
We keep the ones value and carry the tens to the next column.
5 × 4=20+2=22
We keep the ones value and carry the tens value to the next column.
3 × 4=12+2=14
We keep the ones value and carry the tens value to the next column.
1 × 4=4+1=5
So our second row of multiplication is 54280. Notice the 0 holding the ones position open.

	1	3	5	7
×		2	4	6
	8	1	4	2
5	4	2	8	0

Now we multiply 1357 by 200. Since we are multiplying by a multiple of 100, we place a 0 in both the ones and the tens column to hold the position open and multiply by 2.
7 × 2=14
We keep the ones value and carry the tens to the next column.
5 × 2=10+1=11
We keep the ones value and carry the tens to the next column.
3 × 2=6+1=7
We don't have a carry this time.
1 × 2=2
So our third row of multiplication is 271400. Notice the zeros holding the tens and ones positions open.

		1	3	5	7
	×		2	4	6
		8	1	4	2
	5	4	2	8	0
2	7	1	4	0	0

Now we add up the three rows of multiples.

			8	1	4	2
		5	4	2	8	0
+	2	7	1	4	0	0
	3	3	3	8	2	2

So 1357 × 246=333822. That's 1357 added to itself 246 times. We had to make three multiplications and one addition to get the answer. That's a lot faster than doing 246 additions.

Just like with addition, multiplication is commutative; 3 × 2=2 × 3. And, just like with addition, multiplication is also associative;(3 × 2)× 4=3 ×(2 × 4). And, just like with addition, there is also an identity element in multiplication; it is 1. Multiplying a number by 1 doesn't change its identity. This will be an important tool later on. Also, notice that multiplying any number by 0 gives 0. There is an important rule to remember in mathematics that if the product of two numbers is 0, one of them must have been 0 to begin with.

Addition and multiplication can be mixed in a sentence. When we see addition and multiplication together, there is a particular order that we do the operations in. In English, "cats eat birds" means something completely different if we change the order to "birds eat cats." In math, the order of the operations can change the meaning of a mathematical sentence. Look at the sentence 2+3 × 4 for example.What if we add 2 and 3 then multiply by 4. We get 5 × 4=20. But what if we multiply 3 and 4 first, then add 2. We get 2+12=14. In math, multiplication is always done before addition. So 2+3 × 4 is 14. To avoid confusion, we use parenthesis around operations. Operations inside parenthesis are done first. So it would be clear that 2+(3 × 4) is 14 and (2+3)× 4=20.

When mixing multiplication and addition, there is a special property called distribution, when we have multiplication on the outside of a parenthesis containing an addition. In something like
2×(3+4), the multiplication distributes to each term of the addition inside the parenthesis.
2×(3+4) is the same as 2 × 3+2 × 4. It is clear that 2 × 7=14 and 6+8=14. We will use this distributive property a lot in the future.

There is a special multiple that will become important later on. It is called the least common multiple or lcm. It is the smallest number that is a common factor of two numbers. You start with the prime factorization of two numbers, then you count the each factor at least once.

Some rules for factoring numbers are as follows. If the number is even, 2 is a factor. If the number ends in 5, then 5 is a factor, If the sum if the digits of a number add up to a multiple of 3, then 3 is a factor. There are other rules that can be found at Wikipedia http://en.wikipedia.org/wiki/Divisibility_rule

Let's find the least common multiple of 48 and 52. If we just look at the product, 48×52=2496. Now let's factor. The factors of 48 are 2,2,2,2 and 3. The factors of 52 are 2,2, and 13. The lcm of 48 and 52 would be the product of four 2's, a 3 and a 13. 2×2×2×2×3×13=624. The reason the lcm is less than the straight product of 48 and 52 is that we don't have to multiply by the extra 2's in 52; we already counted them in the factors of 48. The smallest number that is a factor of both 48 and 52 is 48×13=624 and 52×12=624.